shall i study integration and then differentiation or should i study differentiation and integration?

Question

what are the concepts should we know to master calculus and understand calculus completely

Answer 1

you have to study differentiation first , later you will study integration becase the base to figure out the integral is differentiation. and sometime you will recognize the solution of that integral by derivative. example take integral(cosx)dx the answer is sinx because you take derivative of sinx you should have cosx which is the answer inside the integral. kinda flip it. if you dont know how to take derivative that is very hard to figure out integral. later on you will see some integral you need to use u substitution which had derivative on it. good luck

Answer 2

Why not do both at the same time? Start with a very smple function, like y = x² , plot it on a piece of paper, and then draw a lot of vertical slices, exactly as if it was a piece of cake and you've sliced it into a lot of thin pieces. From that, and some geometrical thinking, you can:

1) Figure out the slope of short hypotenuse at the top of each slice, and
2) See how the area of the piece of cake increases from left to right piece by piece, the rate of area increase being the slope of that short hypotenuse.

The first is differentiation, the second is integration, and both are connected in this way. Why not look both of them at once at this very simple level, and understand why finding the area under a function is the inverse problem of finding the tangent of that function? Once you firmly understand differentiation and integration for this very simple case, the rest of caculus will be a lot easier.

The normal way "to do business" is to first learn about differentiation, and then follow it with integration, but who cares what educators think? Newton did not "first master differentiation, and then integration", he actually saw both almost at the same time, so why can't you? It's like understanding that multiplication and division are inverses of each other. Every time you learn how to differentiate a function, you can learn its opposite.

Answer 3

Integration is about nano things add up in very large numbers to be eyecatcher. Take an example, take a fixed length and height flight of steps. If the steps are 30 in total, each step has a particular width and height. if we make the height half, keeping the width same, we have to make it 60 number of steps, Go on decreasing the height and / or width of each step as much as you wish, the number of steps rises very fast, ultimately the the steps coincide with the slope line.But the same total height of the steps and the same total height of the steps remain , same as in the original set up. Thus infinite number of nano heights are added up or ' integrated' to get at a fixed height. This is integration.
Differentiation is a process to know how steep the flight of steps are; or the slope of it for that matter. In the above example, The ratio of height to the width of each step is the steepness of the slope, or simply the slope, of the differential co-efficient for that matter. By passing though a step, while covering a certain horizontal distance, we automatically coer a vertical distance too, If the vertical distance is too much in comparison th the horizontal distance, we say the steps are more steep. Ptherwise the slope is less. Suppose we are trekking on a mountain, the slope is not definitely same everywhere. To measure slope at any point, we carve off a small step there and measure the ratio of heght to width. The smaller the step the better the approximation. Thus we compare slopes different points along a contour of a mountain or compare the slopes among different mountains.
What business we do comparing the slopes ? Yes, a lot of , and many kinds of businesses. Guess what. In most natural phenomena, it is easier to observe ' the rate' of change, rather than tracking the phenomenon itself. If only we knew a point, and all the rates of change thereafter, we can reconstruct the graph. Almost entire classical Physics is only this.
Instead of the graph proper, if we knew the rates of change at all points, and knew only one point of the graph, by multiplying small widths to rates of change there, we could well get the small heights at each step and adding it together, we get or are able to reconstruct the entire graph. Now, do the processes differentiation and integration sound reverse processes of each other. Exactly they are so. And integration is still analogous to addition, and differentiation is analogous to division. Of course they are, After all division is repeated subtraction.
In test books conventionally differentiation is taught first and then integration is taught as the reverse process. then it is explained that integration is is process of addition of infinitely many infinitely small elements. On can teach the topics in reverse order of course, for example, I can.But is is easier to stay ith the crowd.
The more inquisitiveness you have, the more fun you derive.

Answer 4

Differentiation ===> Integration

Answer 5

The mechanics behind differentiation and integration (which are inverse operations of each other, as you may already know) aren't hard to learn, but they do take time and practice. Also, as one person above me indicated, you're not behind considering your age. The vast majority of people don't see any calculus whatsoever until 12th grade, and a considerable amount don't it until their freshman year of college.

Answer 6

It actually shouldn't matter once you reach the fundamental theorem of calculus. Still, I would recommend doing derivatives until integrals. It will help things click into place early on, but later on it won't matter because it's a big circle.

Answer 7

I'd begin with differentiation and move on from there. It's easier to understand.

Answer 8

You study diffrentiation first caz integration is inverse operation of differation
exactly like you studied first how to square a number then later how to get square root , they are inverse operations

Answer 9

differentiation then integration

cuz u need to understand derivates b4 going into antiderivates